本文選題：高維單守恒律方程 + 二維格式�。� 參考：《中國(guó)科學(xué)院研究生院(武漢物理與數(shù)學(xué)研究所)》2016年博士論文

【摘要】：本文我們主要研究了二維非線性雙曲守恒律方程的Cauchy解的相關(guān)問(wèn)題。第二章首先介紹了二維單守恒律方程的概念和相關(guān)結(jié)論,然后給出了二維T-C變差和二維有界變差空間的概念和相關(guān)結(jié)論。第三章我們考慮了具有緊支集初值的二維單守恒律方程的Cauchy問(wèn)題,用Riemann解的結(jié)構(gòu)構(gòu)造了一個(gè)二維格式,并最終證明了該格式的極限為熵弱解,其過(guò)程分為以下五個(gè)步驟：在第二節(jié)中,我們時(shí)間以步長(zhǎng)At進(jìn)行分層,在每層開始時(shí),對(duì)初值重新定義,使得其變?yōu)樗钠Ｖ档腞iemann問(wèn)題,然后用Riemann問(wèn)題解來(lái)代表一個(gè)時(shí)間步長(zhǎng)At內(nèi)的解,從而構(gòu)造出二維格式。在第三節(jié)中,我們估計(jì)了該二維格式關(guān)于空間變量x,y的二維T-C變差,利用熵條件以及T-C變差的性質(zhì)證明了該二維格式的T-C變差是一致有界的。在第四節(jié)中,我們考慮該二維格式關(guān)于時(shí)間t的一致連續(xù)性,我們分別討論了在一個(gè)時(shí)間步長(zhǎng)內(nèi)和跨越多個(gè)時(shí)間步長(zhǎng)這兩種情況,并得到了一致的估計(jì)式。在第五節(jié)中,利用第三、四節(jié)的結(jié)論,我們證明了該格式在R2×R+中幾乎處處收斂的意義下趨近于某個(gè)極限函數(shù)u(x,y,t)。在第六節(jié)中,我們證明了該極限函數(shù)u(x,y,t)是滿足方程的熵條件的。由于我們?cè)诿總�(gè)時(shí)間步長(zhǎng)都做了一次小擾動(dòng),我們需要證明這些小擾動(dòng)的控制函數(shù)趨近于O。第四章我們把該二維格式運(yùn)用到求解一類無(wú)界初值uo(x,y)∈Lloc∞(R2)的Cauchy問(wèn)題,這里uo(x, y)局部變差有界且滿足其中r是極坐標(biāo)的半徑。無(wú)界初值和有界初值有著本質(zhì)的區(qū)別,不過(guò)我們還是證明了某種條件下的二維Cauchy問(wèn)題熵弱解的存在唯一性。第五章我們研究了n維非齊次守恒律方程的Cauchy問(wèn)題及奇性解的結(jié)構(gòu)。第一節(jié)介紹了相關(guān)概念及前人的結(jié)果。第二節(jié)研究光滑Cauchy初值的光滑解產(chǎn)生爆破的充分必要條件和爆破時(shí)間,并給出光滑解全局存在的充分必要條件。第三節(jié)我們計(jì)算了兩個(gè)二維非齊次Riemann解的全局結(jié)構(gòu)及其演化。
[Abstract]:In this paper, we mainly study the Cauchy solution of two-dimensional nonlinear hyperbolic conservation law equation.In the second chapter, the concepts and relevant conclusions of two-dimensional simple conservation law equations are introduced, and then the concepts and conclusions of two-dimensional T-C variation and two-dimensional bounded variation spaces are given.In chapter 3, we consider the Cauchy problem of two-dimensional conservation law equation with the initial value of compact set, construct a two-dimensional scheme by using the structure of Riemann solution, and finally prove that the limit of the scheme is entropy weak solution.The process is divided into the following five steps: in the second section, we delaminate with step size at the beginning of each layer, redefine the initial value so that it becomes a four-piece constant Riemann problem.Then the solution of the Riemann problem is used to represent the solution in a time step at, and a two-dimensional scheme is constructed.In the third section, we estimate the two-dimensional T-C variation of the two-dimensional scheme with respect to the spatial variable xy. By using the entropy condition and the properties of the T-C variation, we prove that the T-C variation of the two-dimensional scheme is uniformly bounded.In the fourth section, we consider the uniform continuity of the two-dimensional scheme with respect to time t. We discuss the two cases of a time step size and a span of multiple time steps respectively, and obtain a consistent estimation formula.In the fifth section, by using the conclusions of the third and fourth sections, we prove that the scheme converges to a certain limit function UX ~ XY ~ (t) in the sense of almost everywhere convergence in R ~ 2 脳 R.In the sixth section, we prove that the limit function u _ XY _ t) satisfies the entropy condition of the equation.Since we make a small perturbation at each time step, we need to prove that the control function of these small perturbations approaches O.In chapter 4, we apply this two-dimensional scheme to solve the Cauchy problem for a class of unbounded initial value UOX (y) 鈭，

本文編號(hào)：1763677